TY - JOUR
T1 - Monotone Finite Difference Schemes for Anisotropic Diffusion Problems via Nonnegative Directional Splittings
AU - Cuong Ngo & Weizhang Huang
JO - Communications in Computational Physics
VL - 2
SP - 473
EP - 495
PY - 2018
DA - 2018/04
SN - 19
DO - http://doi.org/10.4208/cicp.280315.140815a
UR - https://global-sci.org/intro/article_detail/cicp/11097.html
KW - 
AB - <p style="text-align: justify;">Nonnegative directional splittings of anisotropic diffusion operators in the
divergence form are investigated. Conditions are established for nonnegative directional
splittings to hold in a neighborhood of an arbitrary interior point. The result
is used to construct monotone finite difference schemes for the boundary value problem
of anisotropic diffusion operators. It is shown that such a monotone scheme can be
constructed if the underlying diffusion matrix is continuous on the closure of the physical
domain and symmetric and uniformly positive definite on the domain, the mesh
spacing is sufficiently small, and the size of finite difference stencil is sufficiently large.
An upper bound for the stencil size is obtained, which is determined completely by the
diffusion matrix. Loosely speaking, the more anisotropic the diffusion matrix is, the
larger stencil is required. An exception is the situation with a strictly diagonally dominant
diffusion matrix where a three-by-three stencil is sufficient for the construction of
a monotone finite difference scheme. Numerical examples are presented to illustrate
the theoretical findings.</p>